Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 36A / b / iii / Solution

Ifthen $e^{\beta a}$ is large and the wall correction is exponentially small:This is exactly the equipartition theorem result from the two kinetic and two harmonic-potential quadratic terms.

If $a\ll k_{B}T$, useThenThe leading term is the kinetic energy. The particle is nearly uniform in the disk, so $⟨r^2⟩\to R^2/2$ and $⟨V⟩\to \kappa R^2/4$, much smaller than $k_{B}T$. The hard wall therefore suppresses the positional equipartition contribution, as in the hard-wall correction to harmonic equipartition.

Solved by gpt-5.6-sol high.